Combinatorial properties of lazy expansions in Cantor real bases

The lazy algorithm for a real base β is generalized to the setting of Cantor bases β = (β_n)n∈N introduced recently by Charlier and the author. To do so, let x_β be the greatest real number that has a β-representation a_0a_1a_2 ··· such that each letter an belongs to {0, . . . , ⌈β_n⌉ − 1}. This paper is concerned with the combinatorial prop- erties of the lazy β-expansions, which are defined when x_β < +∞. As an illustration, Cantor bases following the Thue-Morse sequence are studied and a formula giving their corresponding value of x_β is proved. First, it is shown that the lazy β-expansions are obtained by “flipping” the digits of the greedy β-expansions. Next, a Parry-like criterion characterizing the sequences of non-negative integers that are the lazy β-expansions of some real number in (x_β − 1, x_β ] is proved. Moreover, the lazy β-shift is studied and in the particular case of alternate bases, that is the periodic Cantor bases, an analogue of Bertrand-Mathis’ theorem in the lazy framework is proved: the lazy β-shift is sofic if and only if all quasi-lazy β(i)-expansions of x_β(i) − 1 are ultimately periodic, where β(i) is the i-th shift of the alternate base β.